Method to find optimal BST using DP

1. Design and Analysis of Algorithm(18CS42)
Module 4
Dynamic programming
Lecture Presentation
Divya K S
Dept. of CSE

2. OPTIMAL BINARY SEARCH TREE
• Basic definitions
• Concepts of Optimal Binary Search Tree
• Example Problem
• Assignment

3. BINARY SEARCH TREE
• In A Binary Search Tree, The Value Of All The Nodes In The Left Sub-tree Is Less Than The Value Of The
Root.
• Similarly, Value Of All The Nodes In The Right Sub-tree Is Greater Than Or Equal To The Value Of The
Root.
• This Rule Will Be Recursively Applied To All The Left And Right Sub-trees Of The Root.

4. EXAMPLE

5. KEYS 1, 2,3

6. BST 3BST 2
BST 4
BST 1
BST
5

7. DEPTH=0,HEGHT=1
DEPTH=1,HEIGHT=2
DEPTH=2 HEIGHT=3
HEIGHT*FREQUEN

8. 1*1+2*2+2*6=17

11. Cost: 13

12. • Using dynamic programming find the optimal binary search tree for
the given data.
Keys 10 12 16 21
Frequency 4 2 6 3
0 1 2 3 4
1 0 4
2 0 2
3 0 6
4 0 3
5 0
1 2 3 4
Keys 10 12 16 21
Frequency 4 2 6 3
Solution: Solution matrix

13. Formula:
0 1 2 3 4
1 0 4 8(1)
2 0 2 10(3
)
3 0 6
4 0 3
5 0
C[1,2]=
K=1
C[1,0]+C[2,2]=0+2=2---------
ROOT(1)
K=2
C[1,1]+C[3,2]=4+0=4
+ 6===8
C[2,3]=
I=2
J=3
K=2
C[2,1]+C[3,3]=0+6=6
K=3
C[2,2]+C[4,3]=2+0=2
+8===10

14. 0 1 2 3 4
1 0 4 8(1) 20(3
)
26(3
)
2 0 2 10(3
)
16(3
)
3 0 6 12(3
)
4 0 3
5 0
C[3,4]
k=3
C[3,2]+C[4,4]=0+3=3
i=3,j=4 k=4
C[3,3]+C[5,4]=6+0=6
Frequency=9
=====12ROOT(3)
Minimum cost =2
16
10 21
12 Optimal Binary search
1*6=6
2*4+2*3=14
3*2=6

15. ALGORITHM OPTIMAL BINARY SEARCH TREE

16. ASSIGNMENT